Relativistic Quantum Dynamics: A non-traditional perspective on space, time, particles, fields, and action-at-a-distance
Abstract
This book is an attempt to build a consistent relativistic quantum theory of
interacting particles. In the first part of the book "Quantum electrodynamics"
we follow rather traditional approach to particle physics. Our discussion
proceeds systematically from the principle of relativity and postulates of
quantum measurements to the renormalization in quantum electrodynamics. In the
second part of the book "The quantum theory of particles" this traditional
approach is reexamined. We find that formulas of special relativity should be
modified to take into account particle interactions. We also suggest
reinterpreting quantum field theory in the language of physical "dressed"
particles. This formulation eliminates the need for renormalization and opens
up a new way for studying dynamical and bound state properties of quantum
interacting systems. The developed theory is applied to realistic physical
objects and processes including the hydrogen atom, the decay law of moving
unstable particles, the dynamics of interacting charges, relativistic and
quantum gravitational effects. These results force us to take a fresh look at
some core issues of modern particle theories, in particular, the Minkowski
space-time unification, the role of quantum fields and renormalization and the
alleged impossibility of action-at-a-distance. A new perspective on these
issues is suggested. It can help to solve the old problem of theoretical
physics -- a consistent unification of relativity and quantum mechanics.
Supplementary resource (1)
... Moreover, one can switch to the classical relativistic description by taking the limit ℏ → 0 and considering only states describable by localized quasiclassical wave packets, which can be approximated by points in the phase space. In this limit, observables are replaced by real functions on the phase space, quantum commutators are represented by Poisson brackets, and time evolution is approximated by trajectories in the phase space [27,28]. ...
... Perhaps, the most convincing evidence for the dynamical character of boosts was obtained in the Frascati experiment [55][56][57], which established the superluminal dynamics of the electric field of relativistic charges. This observation was explained from the point of view of the Poincaré-Wigner-Dirac theory in [27,58,59]. ...
In Poincare-Wigner-Dirac theory of relativistic interactions, boosts are dynamical. This means that just like time translations boost transformations have non-trivial eect on internal variables of interacting systems. This is dierent from space translations and rotations, whose actions are always universal, trivial and interaction-independent. Applying this theory to unstable particles viewed from a moving reference frame, we prove that the decay probability cannot be invariant with respect to boosts. Dierent moving observers may see dierent internal compositions of the same unstable particle. Unfortunately, this eect is too small to be noticeable in modern experiments.
... The one particle state can be denoted with momentum p and all collective other quantum numbers σ as |p , σ = a † p ,σ |0 with the vacuum state |0 . Since p can span the entire real line, the Hilbert space of one particle state is infinite dimensional and as a consequence the associated representation can be unitary and the generators of the Poincaré group can be represented by hermitian operators [167] (see [20,[171][172][173] for the details). Furthermore, finding the unitary irreducible representations of the Poincaré group can reduce to finding the unitary irreducible representations of its little group [167]. ...
... As a result, the spin operator originally defined in Eq. This form of the spin operator has been known since early days of quantum mechanics (see [175]) and later it was rederived in different context [172,173,[176][177][178][179]. ...
Tunnel-ionization is investigated in the framework of relativistic quantum mechanics.
For an arbitrary constant electromagnetic field a gauge invariant energy
operator is introduced in order to identify the classically forbidden region for
tunnel-ionization. Furthermore, relativistic features of tunnel-ionization are explored.
A one-dimensional intuitive picture predicts that the ionized electron wave
packet in the relativistic regime experiences a momentum shift along the laser’s
propagation direction. This is shown to be consistent with the well-known strong
field approximation. Furthermore, spin dynamics in tunnel-ionization process
is discussed in the standard as well as in the dressed strong field approximation.
Next, the tunneling time delay is investigated for tunnel-ionization by extending
the definition of the Wigner time delay. Later, this concept is redefined in terms of
the phase of the fixed energy propagator. The developed formalism is applied to
the deep-tunneling and the near-threshold-tunneling regimes. It is shown that in
the latter case signatures of the tunneling time delay can be measurable at remote
distance. Finally, the path-dependent formulation of gauge theory is discussed. It
is demonstrated that this equivalent formulation of gauge theory leads to a canonical
gauge fixing, in which the Feynman path integral becomes more intuitive and
the calculation of the quasiclassical propagator is considerably simplified
... Such a distinct feature of the UCT method makes it useful in covariant calculations of the -matrix either by solving the two-particle Lippmann-Schwinger equation (LSE) for the corresponding -matrix or using the perturbation theory (not obligatorily addressing the Dyson-Feynman expansion). In this context, we would like to note an akin approach developed in [16] to problems of the relativistic QFT that deserves, in our opinion, a very undiverted attention. ...
The method of unitary clothing transformations (UCTs) has been applied to the quantum electrodynamics (QED) by using the clothed particle representation (CPR). Within CPR, the Hamiltonian for interacting electromagnetic and electron-positron fields takes the form in which the interaction operators responsible for such two-particle processes as e−e− → e−e−, e+e+ → e+e+, e−e+ → e−e+, e−e+ → yy, yy → e−e+, ye− → ye−, and ye+ → ye+ are obtained on the same physical footing. These novel interactions include the off-energy-shell and recoil effects (the latter without any expansion in (v/c)2-series) and their on-energy shell matrix elements reproduce the well-known results derived within the perturbation theory based on the Dyson expansion for the S-matrix (in particular, the Møller formula for the e−e−-scattering, the Bhabha formula for e−e+-scattering, and the Klein–Nishina one for the Compton scattering).
... Furthermore, while the whole concept of the vacuum's polarizabilty is based on a view point that is based on the existence of virtual charges, we point out that there are also proposed formalisms based on dressed particle states [31][32][33][34][35] that do not require any virtual or bare particles. It would be very interesting to examine in future work how the vacuum's polarizability would manifest itself in such alternative theoretical frameworks. ...
We examine the accuracy of an intrinsically one-dimensional quantum
electrodynamics to predict accurately the forces and charges of a
three-dimensional system that has a high degree of symmetry and therefore
depends effectively only on a single coordinate. As a test case we analyze two
charged capacitor plates that are infinitely extended along two coordinate
directions. Using the lowest-order fine structure correction to the photon
propagator we compute the vacuum's induced charge polarization density and show
that the force between the charged plates is increased. Although a
one-dimensional theory cannot take the transverse character of the virtual
(force-mediating) photons into account, nevertheless it predicts, in lowest
order of the fine-structure constant, the Coulomb force law between the plates
correctly. However, the quantum correction to the classical result is slightly
different between the 1d and 3d theories with the polarization charge density
induced from the vacuum underestimated by the 1d approach.
... However, in this case the position operator and the notion of localization become dependent on the interaction strength, which is not desirable. An alternative approach is to apply the unitary dressing transformation directly to the Hamiltonian, so that definitions of particles and their observables do not depend on interactions (see section 10.2 in [29]). ...
arXiv:1304.7237. We use a one-dimensional model system to compare the predictions of two different yardsticks to compute the position of a particle from its quantum field theoretical state. Based on the first yardstick (defined by the Newton-Wigner position operator), the spatial density can be arbitrarily narrow, and its time evolution is superluminal for short time intervals. Furthermore, two spatially distant particles might be able to interact with each other outside the light cone, which is manifested by an asymmetric spreading of the spatial density. The second yardstick (defined by the quantum field operator) does not permit localized states, and the time evolution is subluminal.
Within the framework of microscopic three-cluster algebraic models with possible consideration of clustering types (D + n) + Λ, (D + Λ) + n, and (n + Λ) + D, the properties of discrete spectrum states of hypernucleus 4ΛH and continuous spectrum states in the 3H + Λ channel are studied. It is shown that the cluster structure is almost completely determined by the clustering (D + n) + Λ with a rather appreciable effect from the polarization of the binary subsystem (D + n) due to its interaction with the Λ particle.
The gauge freedom in the electromagnetic potentials indicates an underdeterminacy in Maxwell's theory. This underdeterminacy will be found in Maxwell's (1864) original set of equations by means of Helmholtz's (1858) decomposition theorem. Since it concerns only the longitudinal electric field, it is intimately related to charge conservation, on the one hand, and to the transversality of free electromagnetic waves, on the other hand (as will be discussed in Pt. II). Exploiting the concept of Newtonian and Laplacian vector fields, the role of the static longitudinal component of the vector potential being not determined by Maxwell's equations, but important in quantum mechanics (Aharonov-Bohm effect) is elucidated. These results will be exploited in Pt.III for formulating a manifest gauge invariant canonical formulation of Maxwell's theory as input for developing Dirac's (1949) approach to special-relativistic dynamics. © Electronic Journal of Theoretical Physics. All rights reserved.
We study the decay law for a moving unstable particle. The usual
time-dilatation formula states that the decay width for an unstable state
moving with a momentum p and mass M is with being the decay width in the rest frame. In
agreement with previous studies, we show that in the context of QM as well as
QFT this equation is \textit{not} correct provided that the quantum measurement
is performed in a reference frame in which the unstable particle has momentum
p (note, a momentum eigenstate is \textit{not} a velocity eigenstate in QM).
We then give, to our knowledge for the first time, an analytic expression of an
improved formula and we show that the deviation from has a
maximum for but is typically \textit{very} small. Then, the
result can be easily generalized to a momentum wave packet. As a next step, we
show that care is needed when one makes a boost of an unstable state with zero
momentum/velocity: namely, the resulting state has zero overlap with the
elements of the basis of unstable states (it is already decayed!). However,
when considering a spread in velocity, one finds again that
is typically a very good approximation. In the end, it
should be stressed that there is no whatsoever breaking of special relativity,
but as usual in QM, one should specify which kind of measurement on which kind
of state is performed.
Maxwell's (1864) original equations are redundant in their description of charge conservation. In the nowadays used, 'rationalized' Maxwell equations, this redundancy is removed through omitting the continuity equation. Alternatively, one can Helmholtz decompose the original set and omit the longitudinal part of the flux law. This provides at once a natural description of the transversality of free electromagnetic waves and paves the way to eliminate the gauge freedom. Poynting's inclusion of the longitudinal field components in his theorem represents an additional assumption to the Maxwell equations. Further, exploiting the concept of Newtonian and Laplacian vector fields, the role of the static longitudinal component of the vector potential being not determined by Maxwell's equations, but important in quantum mechanics (Aharonov-Bohm effect) is elucidated. Finally, extending Messiah's (1999) description of a gauge invariant canonical momentum, a manifest gauge invariant canonical formulation of Maxwell's theory without imposing any contraints or auxiliary conditions will be proposed as input for Dirac's (1949) approach to special-relativistic dynamics.
We propose an alternative technique for numerically renormalizing quantum field theories based on their Hamiltonian formulation. This method is nonperturbative in nature and, therefore, exact to all orders. It does not require any correlation functions or Feynman diagrams. We illustrate this method for a model Yukawa-like theory describing the interaction of electrons and positrons with model photons in one spatial dimension. We show that, after mass renormalization of the fermionic and bosonic single-particle states, all other states in the Fock space have finite energies, which are independent of the momentum cutoff.
The history of renormalization is reviewed with a critical eye, starting with Lorentz's theory of radiation damping, through perturbative QED with Dyson, Gell-Mann and Low, and others, to Wilson's formulation and Polchinski's functional equation, and applications to "triviality," and dark energy in cosmology.
We study c-morphisms from one Hilbert space lattice (with dimension at least three) to another one; we show that for a c-morphism conserving modular pairs, there exists a linear structure
underlying such a morphism, which enables us to construct explicitly a family of linear maps generating this morphism. As a special case we prove that a unitary c-morphism which preserves the atoms (i.e. maps onedimensional subspaces into one-dimensional subspaces) is necessarily an isomorphism. Counterexamples are given when the Hilbert space has dimension 2.
Quantum field theory is a powerful language for the description of the subatomic constituents of the physical world and the laws and principles that govern them. This book contains up-to-date in-depth analyses, by a group of eminent physicists and philosophers of science, of our present understanding of its conceptual foundations, of the reasons why this understanding has to be revised so that the theory can go further, and of possible directions in which revisions may be promising and productive. These analyses will be of interest to graduate students and research workers in physics who want to know about the foundational problems of their subject. The book will also be of interest to professional philosophers, historians and sociologists of science, because it contains much material for metaphysical and methodological reflections, for historical and cultural analyses, and for sociological analyses of the way in which various factors contribute to the way the foundations are revised.
The existence of a unitary representation of the inhomogeneous Lorentz group, which corresponds to a relativistic version of a modified Lee model, is proved through the process of actual construction of set of ten generators (Pµ, Mµν) in terms of creation- and annihilation-operators of three kinds of particles. The model theory contains ab initio a cutoff form factor which is kept, throughout the whole scheme of the formalism, as it stands. The cutoff from factor is a function only of the invariant square of momentum transfer at the primary interaction vertex. By virtue of the cutoff form factor the theory has no divergence difficulties. The total Hamiltonian, H(≡-iP4), of the model contains, from the very beginning and in an inevitable way, a finite “mass-renormalization term”. The inclusion of that term in the Hamiltonian is inevitable in order for the ten generators to satisfy the set of fundamental commutator equations. Thus it turns out that the “mass-renormalization term” subtly takes its proper place within the realm of the formalism.
This chapter discusses the concepts of quantum mechanics, complex Hilbert space, the Lattice structure of general quantum mechanics, the group of automorphisms in a proposition system, and projective representation of the Poincaré group in quaternionic Hilbert space. Theoretical physics in the first half of the 20th century is dominated by two major developments—the discovery of the theory of relativity and the discovery of quantum mechanics. Both have led to profound modifications of basic concepts. Relativity in its special form has proclaimed the invariance of physical laws with respect to Lorentz transformations and led to the inevitable consequence of the relativity of spatial and temporal relationships. On the other hand, quantum mechanics recognizes as basic the complementarity of certain measurable quantities for microsystems and the concomitant indeterminism of physical measurements. From the mathematical point of view, the central object in the special theory of relativity is a group, the Lorentz group, or more generally, the Poincaré group. For quantum mechanics, the most important mathematical object is the Hilbert space and its linear operators.
IN SOME QUARTERS, AT LEAST, IT COUNTS as the “received view” that there cannot be a relativistic, quantum mechanical theory of (localizable) particles. In the attempt to reconcile quantum mechanics with relativity theory, that is, one is driven to a field theory; all talk about “particles” has to be understood, at least in principle, as talk about the properties of, and interactions among, quantized fields. I want to suggest, today, that it is possible to capture this thesis in a convincing “no-go theorem”. Indeed, it seems to me that various technical results on the “non-localizability” of particles in (so-called) relativistic quantum mechanics, going back some thirty years, are best understood as versions of such a theorem.1